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<h2>merge.c -- facet merge operations</h2>
<blockquote>
<p>Qhull handles precision problems by merged facets or joggled input.
Except for redundant vertices, it corrects a problem by
merging two facets. When done, all facets are clearly
convex. See <a href="../html/qh-impre.htm">Imprecision in Qhull</a>
for further information. </p>
<p>Users may joggle the input ('<a href="../html/qh-optq.htm#QJn">QJn</a>')
instead of merging facets. </p>
<p>Qhull detects and corrects the following problems: </p>
<ul>
<li><b>More than two facets meeting at a ridge. </b>When
Qhull creates facets, it creates an even number
of facets for each ridge. A convex hull always
has two facets for each ridge. More than two
facets may be created if non-adjacent facets
share a vertex. This is called a <em>duplicate
ridge</em>. In 2-d, a duplicate ridge would
create a loop of facets. </li>
</ul>
<ul>
<li><b>A facet contained in another facet. </b>Facet
merging may leave all vertices of one facet as a
subset of the vertices of another facet. This is
called a <em>redundant facet</em>. </li>
</ul>
<ul>
<li><b>A facet with fewer than three neighbors. </b>Facet
merging may leave a facet with one or two
neighbors. This is called a <em>degenerate facet</em>.
</li>
</ul>
<ul>
<li><b>A facet with flipped orientation. </b>A
facet's hyperplane may define a halfspace that
does not include the interior point.This is
called a <em>flipped facet</em>. </li>
</ul>
<ul>
<li><strong>A coplanar horizon facet.</strong> A
newly processed point may be coplanar with an
horizon facet. Qhull creates a new facet without
a hyperplane. It links new facets for the same
horizon facet together. This is called a <em>samecycle</em>.
The new facet or samecycle is merged into the
horizon facet. </li>
</ul>
<ul>
<li><b>Concave facets. </b>A facet's centrum may be
above a neighboring facet. If so, the facets meet
at a concave angle. </li>
</ul>
<ul>
<li><b>Coplanar facets. </b>A facet's centrum may be
coplanar with a neighboring facet (i.e., it is
neither clearly below nor clearly above the
facet's hyperplane). Qhull removes coplanar
facets in independent sets sorted by angle.</li>
</ul>
<ul>
<li><b>Redundant vertex. </b>A vertex may have fewer
than three neighboring facets. If so, it is
redundant and may be renamed to an adjacent
vertex without changing the topological
structure.This is called a <em>redundant vertex</em>.
</li>
</ul>
</blockquote>
<p><b>Copyright © 1995-2003 The Geometry Center, Minneapolis MN</b></p>
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